Terminology for exponentiated coefficients from GLMs without a time offset In GLM count models with offsets for time, I know that the exponentiated coefficient is commonly referred to as an incidence rate ratio, which makes sense given that the quantity being modeled is a count over a time period. In GLM count models without a time offset or GLM models for continuous outcomes (e.g., gamma or Tweedie models), however, I have seen much less uniformity in what the appropriate term for the exponentiated coefficient is, and I am trying to figure out what terminology to use to describe my results. Specifically, I have used a zero-truncated negative binomial model without a time offset for hospital length of stay (in days) and a Tweedie model for health expenditures. References I can find in the literature on these models indicate that I might describe exponentiated coefficients from both of these models as "rate ratios." I feel fine about the "ratio" part, as I understand how to interpret these coefficients, but the "rate" part doesn't seem to be accurate. Is there a consensus about the appropriate terminology for exponentiated GLM coefficients from models in which time is not a factor?
 A: One way to think about a GLM model without an offset is that it models a transformation of the expected value of the response variable as a function of the predictor variables included in the model.  (Synonims for the word expected include mean and average.)  The transformation being used depends on the link function used when fitting the model.
If the GLM model is a Poisson model with a log link, then you are modelling the log expected value of the response variable as a function of the predictor variables.  The response variable is a count variable.
As an example, let’s say that your response variable is the number of salmon returning to their spawning ground and that you record the values of this variable each year for 20 years.  Your Poisson model includes a single predictor, year. The model assumes that year has a linear effect on the log expected number of salmon returning to their spawing ground:
$log(expected number) = \beta_0 + \beta_1 year$
if you exponentiate the coefficient $\beta_1$, you get the multiplicative factor by which the expected number of salmon returning to their spawing ground changes for each additional year.  If you estimate $\beta_1$ to be 0.9 from the data, that means that expected number of salmon returning to their spawing ground is estimated to decrease by 10% for each additional year.
For a gamma model with a log link, you would be modelling the log expected value of a (strictly positive) continuous response variable but the interpretation of the regression coefficient for a predictor would be similar to what I described above.
Personally, I find it helpful to frame the interpretation of these types of models in terms of the expected value of the response variable. This way, you know exactly what you are modelling, rather than using mysterious labels that may not even be appropriate for your context.
Going back to the Poisson model above, let’s say that each year you record those salmon counts, you have to account for the fact that the counting effort spanned different numbers of counting days (e.g., only 20 days were used to count fish in 2019, compared to 30 days in 2020). Your model could include an offset, which will be given by
log(counting days):
$log(expected number) = log(counting days) + \beta_0 + \beta_1 year$
But this is the same as:
$log(expected number/counting days) = \beta_0 + \beta_1 year$
so you can see that what we are now modelling is the log-transformed expected number of salmon returning to their spawning ground per counting day. Thus, $exp(\beta_1)$ is the multiplicative factor by which the expected number of salmon returning to their spawning ground per counting day changes for each additional year.
